The first three that come to mind are: Heinlein's "Time for the Stars", Haldeman's "The Forever War", and Anderson's "Tau Zero".
From those titles, a DDG search finds http://sf-encyclopedia.com/entry/relativity listing more, also containing the line "Very many sf stories use relativistic time dilation for one-way Time Travel into the future."
There are very few sci-fi movies or books where relativity is important and they treat it seriously. Tau Zero is one, I'm not sure about Time for the Stars, but Forever War does have FTL.
I created a list of all these works I could find a few years ago. Here is one link:
Virtually all of Larry Niven's novels that don't have FTL touch on this. Also shows up in most of Alastair Reynolds' novels, and in the Three Body Problem series.
alastair reynold's revelation space series certainly doesn't have ftl, apart from maybe [jumper clowns spoiler deleted]; the conjoiner drives use [science] to achieve 1g acceleration to get to near light speed, and time dilation is part of many of the plots.
> the conjoiner drives use [science] to achieve 1g acceleration to get to near light speed
For [science] read [magic], more or less, but yeah. In the very first stories they were Bussard ramjets, but this got retconned out (and a Bussard ramjet actually shows up in a later book as a failed experiment). In most of the books they're more or less applied magic, though.
This, incidentally, seems to be a common theme, as later discoveries tended to fall down on the side of Bussard ramjets not working (due to insufficient density etc). The last of Niven's Known Space books have some special pleading for how the pilot has to carefully direct the ramjet to get sufficient combustion volume, a detail that was never present in the old ones.
Time for the Stars has FTL at the end. (Physical FTL, that is. Telepathic FTL exists at the beginning.) But it's otherwise it's deliberately structured around the twin paradox - the main character is even a twin.
Indeed, while I remembered the importance of time dialation in Forever War , I didn't remember the FTL travel between collapsars.
I'm sure it has been, but I'm mainly talking about popular sci-fi movies. I'm sure you have at least one in mind where they have these 'sleep' devices.
I'm not much into visual media, so I can only come up with a few examples of 'sleep' devices:
Star Trek's "Space Seed" - 200 years in sublight stasis, no mention of speed or distance. Could be 0.5c for all we know. 100 ly at Enterprise's warp 6 cruising speed would take about 4 months, which seems not unreasonable.
The movie "Alien" - stasis, but seemingly with FTL given the times and distances involved. (https://avp.fandom.com/wiki/LV-223 says it was a 2 year voyage to the moon LV-223 in Prometheus.)
What came to my mind immediately was 'Passengers'. But, reading up the plot on Wikipedia, it is possible that it never actually was mentioned how far away the destination was or how fast they were going. So, maybe it wasn't a misconception on the sci-fi side, but on my side. For sure, any light speed travel shouldn't take any time and infinite amount of time should pass for the rest of the universe.
I mean yes, but there is a rather specific window where they make sense: at ~1-80% of c. Lower speeds it doesn't make sense to engage in interstellar travel at all, because even travelling 100s of years won't get you anywhere. At higher speeds it doesn't make sense because of time dilation.
But that's a... pretty important window. There are not-completely-bonkers hypothetical spacecraft designs that could get to, say, 0.2c. There's nothing particularly plausible that could get to 0.99c.
The first three that come to mind are: Heinlein's "Time for the Stars", Haldeman's "The Forever War", and Anderson's "Tau Zero".
From those titles, a DDG search finds http://sf-encyclopedia.com/entry/relativity listing more, also containing the line "Very many sf stories use relativistic time dilation for one-way Time Travel into the future."